Composite Functions
When two functions are combined in such a way that the output of one function becomes the input to another function, then this is referred to as composite function.
Consider three sets X, Y and Z and let f : X → Y and g: Y → Z.
According to this, under map f, an element x ∈ X is mapped to an element y = f(x) ∈ Y which in turn is mapped by g to an element z ∈ Z in such a manner that z = g(y) = g[(f(x)] .
This mapping comprising of mappings f and g is known as composition of mappings. It is denoted by gof . Therefore, we are mapping onto .
The composite function is denoted as:
\(~~~~~~~~~\)
Similarly, (fog) (x) = f (g(x))
So, to find (gof) (x), take f(x) as argument for the function g.
Let us try to solve some questions based on composite functions.
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Let’s Work Out: Example: Given the function f(x) = 3x + 5 and g(x) = \( 2x^3 \) Solution: We know, (gof) (x) = g(f(x)) = g (3x+5) = \( 2(3x+5)^3 \) Using Binomial Expansion, we have \((gof) (x) = 2 \left [ (3x)^{3} + 3.(3x)^{2}.5 + 3.(3x)(5)^{2}+(5)^3 \right ]\) \(\Rightarrow (gof) (x) = 2 \left [ 27x^{3} + 135x^{2} + 225 x + 125 \right]\) \(\Rightarrow (gof) (x) = 54x^{3} + 270x^{2} + 450 x + 250 \) Now, (fog)(x) = f \(\left( g(x) \right)\) \(\Rightarrow (fog)(x) = 6x^{3} + 5\) Example: Let f(x) = \(x^2\) Solution: (gof)(x) = g( f(x)) = g(\(x^2\) (fog) (x) = f(g(x)) = f( \( \sqrt{1 – x^2} \) |
Inverse of a Function
Let f:X → Y. Now, let f represent a one to one function and y be any element of Y , there exists a unique element x ∈ X such that y = f(x).Then the map
\(~~~~~~~~~~\)
which associates to each element is called as the inverse map of f.
The function f(x) = \( x^5 \)
\( f\left( g(x) \right) \)
\( g\left( f(x) \right) \)
Thus, if two functions f and g satisfy \( f \left( g(x) \right) \)
For finding the inverse of a function,we write down the function y as a function of x i.e. y = f(x) and then solve for x as a function of y.
To have a better insight on the topic let us go through some examples.
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Let’s Work Out: Example: If f(x) = \(x^2\) Solution: \( h\left( g(x) \right) \) fohog(x) = f \( \left[ h\left(g(x)\right)\right] \) This is the required solution. Example: Example 2: Find the inverse of the function f(x) = \( x^3 \) Solution: The given function f(x) = \( x^3 \) To find the inverse, we need to write down this function as \(~~~~~~~~~~~~~\) In the above equation,y is an arbitrary element from the range of f. If we solve for x from the above equation, we will get: \(~~~~~~~~~~~~~\) This gives a function g:Y →X. This new function g can be defined as \(~~~~~~~~~~~~~\) This function g is the inverse of the function f since it’s domain is same as the range of the function f.Since g(y) = \( y ^{\frac 12} \) |
So, wasn’t that easy and fun? We have learnt about composition of functions and inverse functions. To learn more about functions, please visit our website www.byjus.com or download our app-BYJU’s the learning app. Happy Learning!!
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